Algorithms for Flexible Equalisation in Wireless Communications
R. Perry, D. R. Bull and A. Nix
Centre for Communications Research
University of Bristol, Bristol, BS8 1TR, UK.
Tel: +44 1 i7 928 7740
Abstract : In this paper the requirements for a flexible
equaliser architecture for wireless communications are
discussed and illustrated with results from simulation.
Decision Feedback Equalisers are compared in terms of their
performance and computational requirements. It has been
found that the recursive modified Gram Schmidt algorithm
provides superior BER performance to the least squares
lattice decision feedback lattice and the conventional RLS
algorithm. In addition the problem of synchronising the
equaliser to achieve best performance and exploit the
multipath propagation of the wideband channel is addressed.
For a two path Rayleigh fading channel model it is shown that
best performance is obtained by synchronising the frame to
the arrival time of the first multipath rather than the
dominant path.
I. Introduction
There are several major TDMA systems currently in
operation or being deployed throughout the world [ 11. In
most systems equalisation is required to maintain
acceptable perfomance. With the development of future
generations of universal mobile communication systems
(UMTS/MBS) there is likely to be significant interest in a
flexible equaliser architecture which can be configured to
compensate for highly variant channel conditions. In this
paper the issues concerning the choice of algorithm for
such a flexible equaliser are discussed. The training data in
current TDMA systems is located at either the start (a
preamble) or centre (a midamble) of a transmitted packet. It
is proposed therefore that the equaliser is trained offline by
buffering each received frame [7]. This allows symbol
timing and frame synchronisation to be determined prior to
training. The choice of equaliser length and exponential
weighting factor required for starting up the DFE are
described in section 11. Due to the very large time delay
spread of the new ETSI Hiperlan standard [7], a low
complexity algorithm is proposed for this system in section
11. The issue of frame synchronisation is discussed in
section 111.
11. Equaliser Algorithms
Decision Feedback Equalisers (figl) trained by the
recursive least squares (RLS) algorithm are chosen because
of their fast convergence and relatively low complexity [2].
In addition, the basic structure of the equaliser is
Fax: +44 117 925 5265
t t t
I
Figure 1 Transversal DW
independent of system type and it is therefore relatively
simple to reconfigure the equaliser for each system.
Important factors that affect the performance and determine
how the equaliser is set up are the equaliser length and the
exponential weighting factor applied to the input data. The
length can be selected from a channel impulse response
(CIR) estimate derived from the synchronisation sequence.
The clocking rate must be adjusted to deal with the
different data rates. The appropriate training sequence must
also be supplied, together with the modulation scheme for
decision directed feedback. The four leastsquare
algorithms considered here (apriori forms) are the Direct
form RLS, Complex Fast Kalman Algorithms based on
transversal filter structures [2,4], the Lattice Least Squares
Decision Feedback equaliser [5] and the Recursive
Modified Gram Schmidt algorithm [6].
11.1 Simulation Study
For the study, simulated GSM and NADC(1S54) system
models have been developed based on the
recommendations in [3] and [4]. In the NADC simulation,
root raised cosine filtering at the receiver and transmitter (a
=0.35) is used. The use of both d4DQPSK and QPSK has
been considered. For the GSM system, Gaussian pulse
shaping in the transmitter is used. At the receiver a
Butterworth five pole filter was used with 3dB bandwidth
0.5D [21. The results obtained using QPSK modulation are
shown in fig 2 for a two path channel with independently
faded paths of equal mean power. For each frame a new
random channel realisation was used. The performance of
the RMGS algorithm was found to be marginally better
than the LSL and DRLS algorithms. The use of decision
0780325702/95 $4.00 01995 IEEE 1940
directed update was found eo cause a significant loss in
performance.
~ ~
IFI
3E2
b w E?
35.3
E3
3E4
m
Yl

.
I
4 6 8 IO 12 14 16 18 20 1E4' ' '
Eb/No(dB)
Figure 2: Comparative perfonnance of DFE equalisers at 20mph over a
two path channel.
11.2 Equaliser Time Span and Complexity
The computational complexity of any algorithm is
dependent on the required equaliser length. For the NADC
and JBC systems the time dispersion covers at most two
symbol periods so a short equaliser can be used. In [4] it
has been shown that for IS1 spanning less than a third of a
symbol period an equaliser will degrade performance and
should therefore be switched out. The complexity of the
algorithms are compared below based on the number of
feedforward (N 1) and feedback (N2) stages, where
N=Nl+N2 is the total number of stages. For the CFKA p=2
or 3 depending on whether fractionally or symbol spaced
feedforward taps are used.
NADC and DECT systems these fast algorithms may not be
the most appropriate choice. However, the modularity of
the LSL and RMGS algorithms can be exploited to increase
processing speed which is particularly important for high
speed systems. The length of the equaliser is also restricted
by the amount of training data provided and therefore
relatively short equalisers must be used. The complexity of
the DRLS and RMGS algorithms are therefore comparable
to the fast algorithms. However the need for fractionally
spaced feedforward taps will increase the number of
equaliser taps.
11.3 Channel Tracking
Channel time variation requires the use of tracking. The
GSM system has specified the use of a midamble training
sequence which minimises the channel variation across the
packet. For the NADC system the longer frame duration
resu!ts in significant time variation. For a DFE the choice
of the tracking parameter is critical and must be chosen to
balance the algorithm noise introduced and the need to
track rapid channel variations. Figure 3 shows the
performance of the RMGS algorithm for the NADC
simulation (50rnph) as a function of the weighting factor
and EbfiTo. At low values of Eb/No a low value of
weighting factor introduces an unaccepeable level of
additional noise and so larger values are required. This is
true of the other algorithms, but it was found that the
optimum values for the weighting factor in these algorithms
were generally larger. The results indicate that performance
degrades rapidly when channel tracking is not used and
therefore the tracking parameter should generally be kept
small if the rate of change of the channel is unknown.
(1) LMSDFB, (2)DRLS, (3)CFKA (4)LSLDFE (5) RMGS, (6)Cholesky
1 E4
5E1
2E1
1 E1 U
2E2
Decomposition (ah requires N square roots)
Table 1:Relative Complexity of DFE Algorithms
The Cholesky decomposition niethod in Table 1 is used to
compute the DFB tap weights of a transversal filter directly
from the Wiener Hopf equations. The relatively high
computational count for this approach is offset by the fact
that this operation is carried out only once for the packet. It
is therefore most appropriate for quasi static channels when
channel tracking is not required. The computational flow is
also less regular and might make implementation difficult.
The fast RLS algorithms only provide significant
computational savings when the equaliser length is greater
than five symbols [21. For the short CIRs occurring on the
2E3 I
1E3 I
0.7 0.75 08 OB 09 095 1
Weighting Factor
Figure 3: Effect of varying weighting factor as a function of Eb/Nc
11.4 Combined LMS/CI Estimation Equaliser
The proposed Hiperlan standard [7] is being developed to
support data rates of approximately 20Mb/s. At this high
rate the ISI is predicted to cover up to seven symbol
periods. An implementation of the Square Root Kalman
algorithm for use on this system is currently being
developed but requires the use of four C40 processors in
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parallel [7]. An alternative approach is to compute the
equaliser tap weights directly from the WienerHopf
equations formed from an estimate of the CIR. For
illustration, the performance of this type of approach is
compared with the DRLS algorithm over a simplified form
of the model adopted in [9].
The approach adopted here is to use the solution derived
from the WienerHopf equations (without noise estimation)
to initialise the tap weights of the equaliser. The equaliser
is then fine adjusted using the LMS algorithm, by reusing
the training data. This approach does not require any
assumptions regarding the correlation of the interfering
noise. The WienerHopf equations were solved using a
Cholesky factorisation of the data correlation matrix. A 48
symbol synchronisation sequence was used as in [9] to
derive the channel estimate and carry out frame
synchronisation The performance of the proposed method
is compared with the conventional RLS algorithm in fig 4
over a channel with a RMS delay spread of approximately
2011s. The effect of varying the excess bandwidth of the
receive filtering has also been varied.
1 E+O
3E1
1 E1
3E2
CT
W 1E2
3E3
1 E3
3E4
1 E4
m
0 5 10 15 20
Eb/No(dB)
Figtire 4: Comparison of DFEs over a stationary channel
The difference in performance between the DRLS
algorithm and LMS based algorithm is very slight. This
could be due to the slightly larger excess mean square
inherent in the LMS algorithm. However convergence of
the LMS algorithm was possible within the given training
length. The comparatively low complexity of this type of
approach over the DRLS algorithm makes it attractive for
the Hiperlan system. This technique can also be applied to
the other systems at reasonable computational expense,
provided channel variations are not too rapid.
111. Synchronisation
The issue of frame synchronisation is described in this
section. A 2Path channel model that has been used
extensively in previous simulation studies [IO] is
considered. In general, the CIR is represented by the
complex transfer function (for multipaths separated by a
symbol period)
where hi is a Rayleigh fading process. A real CIR will be
considered here for ease. It is assumed that the transmitted
data and channel multipaths are uncorrelated. For a
minimum phase impulse response the optimum equaliser
coefficients are given by
H( z)=ho+h,z' (1)
where F, , k = 0.. N, and B, k = 1.. N, are the feedforward
and feedback equaliser taps respectively (Figure 1). For a
non minimum phase channel, the equaliser tap weights are
dependent on the burst timing adopted. In [lo] it was
assumed that the received sample at time t from which an
estimate of d(t) is made is given by
where q(t) is additive Gaussian noise. In this case the IS1 is
entirely precursor and the feedback filter is not used. By
fixing the starting point at the arrival time of the first
multipath, the IS1 will now be entirely post cursor and the
decision feedback effect is obtained. The sample u(t+l)
contains the desired symbol as well and therefore the
feedforward filter contains signal energy from the two
multipaths. In [ 101 because of the timing adopted, a variable
reference tap was used, which was set at the penultimate
feedforward filter tap. This has the same effect as sampling
the first arriving ray. For the non minimum phase CIR, the
equaliser tap values are given by [8]
u(t) = h,d(t+l)+h,d(t)+q(t) (3)
This solution in terms of the feedback tap has also been
obtained from the WienerHopf equations. The exact
solution in [8] was obtained by choosing B, to minimise the
noise power at the output of the equaliser. The power
spectral density of the noise is No and therefore the noise
power at the equaliser output is
where 1 H2
The total noise is minimised when
(5) 1
B,' +(~HB,)'G
H = ho / h,
This solution assumes that the residual IS1 is negligible and
that noise is present. It has been found that as the noise
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power becomes vanishingly small. in order to minimise the
residual ISI, only the feedforward filter tap F, will be non
zero. To account for this, the residual IS1 power contributed
by the last feedforward tap is included in (5) which leads to
a modified solution for 13,.
HGN, +lb12~~~f
B, = (7) N, + H~GN, +
To compare performance for minimum and non minimum
phase channels, the desired signal energy (obtained from
the weighted sum of the components of the desired symbol
in the input samples contained in the feedforward section) is
compared to the total interference (noise and residual ISI).
Assuming llio12 + Ih, l2 = 1 and an Eb/No value of 20dB the
ratio of the power in the desired symbol to the interfering
power as a function of the relative power in the two paths is
shown in Fig 5.
Min Phase Max Phase
30 m 10 0 IO 11
IbllmtLw
Figrare 5: Effect of Channel Impulse Response on DFE performance
As the feedforward filter length is increased the DFEs
performance for a non minimum phase CIR approaches that
for the minimum phase channel. This is verified by
allowing N, to tend to infinity in (5). In addition, as H+
1, B, + 1 and all the feedforward filter taps apart from the
first tend to zero, which is consistent with the minimum
phase solution (2). A time reversal equaliser uses the energy
from whichever path is currently dominant and can
therefore achieve the same perfomance, for both minimum
and non minimum phase channels, but with fewer taps.
If the dominant path in a non minimum phase CIR is used
for synchronisation, only the feedforward filter is used and
therefore, the noise power at the equaliser output is
approximated by (ZF criterion)
by ~~1Q~~~g the equaliser output will.
reduce the perfomaxe penalty.
In this paper various. algorithms have been considered for
the training of a flexible equaliser. DEE type structures
have been selected because of their lower computational
complexity relative to MLSE. algorithms. The results shown
here indicate that the RMGS algorithm provides marginally
superior BER performance to both the DRLS algorithm and
LSL DFE. In addition it is capable of tracking very fast
time varying channels. Its complexity is similar to the
DRLS algorithm and for short time spans its computational
requirements compared to the LSL are favourable. It also
shares with the LSL a high degree of modularity which is
attractive for high speed applications. The RMGS
algorithm is therefore a potentially attractive solution for a
flexible architecture. The importance of frame
synchronisation and its effect on the performance of the
DRLS DFE was also highlighted. For a two path channel,
synchronising the frame to the first arriving ray reduces the
total noise power at the equaliser. output compared to
synchronising to the dominant ray.
V. Acknowledgements
The authors wish to express their gratitude to the members
of the Centre for Communications Research, University of
Bristol. We also wish to thank EPSRC and Hewlett Packard
Laboratories for their support of this work.
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EEE PIMRC 1994 ~013
As H+l the total noise power tends to No(Nr + I)// hlI2
which is clearly larger than in (5). Relaxing the ZF criterion
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